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# Inverse Variation Models

## Identify and solve y=k/x form equations

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Inverse Variation

The force, F, required for a karate student to break a board varies inversely with the board's length, L. It takes 21 pounds of pressure to break a board that is 3 feet long. How many pounds of pressure does it take to break a board that is 2 feet long?

### Inverse Variation

We say that a set of data is related inversely if the independent increases and dependent variables decreases or vice versa. For example, the further away from an object that you are, the smaller it appears. In inverse variation, the variables are related inversely. As \begin{align*}x\end{align*} gets bigger, \begin{align*}y\end{align*} would get smaller. The inverse variation equation is \begin{align*}y=\frac{k}{x};k,x \ne 0\end{align*}. We still call \begin{align*}k\end{align*} the constant of variation and \begin{align*}y\end{align*} is said to vary inversely with \begin{align*}x\end{align*}. \begin{align*}k\end{align*} can also be written \begin{align*}k=xy\end{align*}.

The variables \begin{align*}x\end{align*} and \begin{align*}y\end{align*} vary inversely, and \begin{align*}y=7\end{align*} when \begin{align*}x=2\end{align*}. Let's write an equation that relates \begin{align*}x\end{align*} and \begin{align*}y\end{align*} and find \begin{align*}y\end{align*} when \begin{align*}x=-6\end{align*}.

Using the inverse variation equation, we can substitute in \begin{align*}x\end{align*} and \begin{align*}y\end{align*} and solve for \begin{align*}k\end{align*}.

\begin{align*}k&=yx \\ k&=7 \cdot 2 \\ k&=14\end{align*}

Therefore, the equation is \begin{align*}y=\frac{14}{x}\end{align*}. To find \begin{align*}y\end{align*} when \begin{align*}x\end{align*} is -6, we have \begin{align*}y=\frac{14}{-6}=- \frac{7}{3}\end{align*}.

Now, let's determine if the following set of data varies directly, inversely, or neither and find the equation if possible.

\begin{align*}x\end{align*} 1 2 3 4
\begin{align*}y\end{align*} 12 6 4 3

Looking at the set of data, the \begin{align*}x\end{align*} values increase. For the data to vary directly, the \begin{align*}y\end{align*} values would also have to increase, and they do not. So, it could be an inverse relationship. Let’s see if \begin{align*}k\end{align*} is the same for every set of points.

\begin{align*}k&=1 \cdot 12=12 \\ k&=2 \cdot 6 \ =12 \\ k&=3 \cdot 4 \ =12 \\ k&=4 \cdot 3 \ =12\end{align*}

So, for each set of points, \begin{align*}k = 12\end{align*}. Therefore, the equation is \begin{align*}y=\frac{12}{x}\end{align*}. If \begin{align*}k\end{align*} is not the same, then the answer would have been neither.

Finally, let's solve the following problem.

Sherry is driving from San Francisco to Los Angeles (380 miles). How long does it take her if she drives 65 miles per hour (the speed limit)? How fast does she have to drive to get to LA in five and a half hours?

The faster Sherry drives, the less time it will take her to get to LA. Therefore, this is an inverse relationship. \begin{align*}y\end{align*} is the time driving, \begin{align*}k\end{align*} is the 380 miles between LA and San Francisco and \begin{align*}x\end{align*} is the speed.

\begin{align*}y=\frac{380}{x}\end{align*}

So, it is going to take her \begin{align*}y=\frac{380}{65} \approx 5.85\end{align*} hours, which is 5 hours and 51 minutes. For her to get there in 5.5 hours, she would have to drive \begin{align*}5.5=\frac{380}{x} \rightarrow 5.5x=380 \rightarrow x=69.1\end{align*} miles per hour.

### Examples

#### Example 1

Earlier, you were asked to determine the pounds of pressure it takes to break a board that is 2 feet long.

We are told this is an inverse variation, so we can use the inverse variation equation \begin{align*}y=\frac{k}{x}\end{align*}. In this case, y equals the force and x equals the length of the board.

\begin{align*}F = \frac{k}{L}\\ 21 = \frac{k}{3}\\ k = 63\end{align*}

We've found the constant of variation, so now we use the equation a second time to find the force when the length of the board is 2 feet.

\begin{align*}F = \frac{63}{2}\\ F = 31.5\end{align*}

Therefore 31.5 pounds of pressure are needed to break a board that is 2 feet long.

#### Example 2

The variables \begin{align*}x\end{align*} and \begin{align*}y\end{align*} vary inversely, and \begin{align*}y=-5\end{align*} when \begin{align*}x = 3\end{align*}. Find the equation and determine \begin{align*}x\end{align*} when \begin{align*}y = 12\end{align*}.

First, solve for \begin{align*}k\end{align*}.

\begin{align*}k=xy \rightarrow 3 \cdot -5&=-15 \\ y&=- \frac{15}{x}\end{align*}

Now, substitute in 12 for \begin{align*}y\end{align*} and solve for \begin{align*}x\end{align*}.

\begin{align*}12&=- \frac{15}{x} \\ 12x&=-15 \\ x&=- \frac{5}{4}\end{align*}

#### Example 3

Determine if the set below varies directly or inversely.

\begin{align*}x\end{align*} 1 2 3 4 5
\begin{align*}y\end{align*} 2 6 12 24 36

At first glance, it looks like both values increase together, so we know the set does not vary inversely. Let’s check for direct variation by determining if \begin{align*}k\end{align*} is the same for each set of points.

\begin{align*}k=xy=2 \ne 3 \ne 4 \ldots\end{align*}

None of these points have the same ratio; therefore the data set does not vary inversely or directly.

#### Example 4

It takes one worker 12 hours to complete a specific job. If two workers do the same job, it takes them 6 hours to finish the job. What type of relationship is this? How long would it take 6 workers to do the same job?

This is an inverse relationship because as the number of workers goes up, the number of hours it takes to complete the job goes down. \begin{align*}k=12 \cdot 1=2 \cdot 6=12\end{align*} and the inverse variation equation is \begin{align*}y=\frac{12}{x}\end{align*}. For 6 workers to complete the job, it would take \begin{align*}y=\frac{12}{6}=2 \ hours\end{align*}.

### Review

For problems 1-4, the variable \begin{align*}x\end{align*} and \begin{align*}y\end{align*} vary inversely. Use the given \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values to write an inverse variation equation and find \begin{align*}y\end{align*} given that \begin{align*}x =15\end{align*}.

1. \begin{align*}x=4,y=3\end{align*}
2. \begin{align*}x=\frac{1}{5},y=10\end{align*}
3. \begin{align*}x=8,y=\frac{3}{4}\end{align*}
4. \begin{align*}x=\frac{2}{3},y=\frac{15}{8}\end{align*}

For problems 5-8, the variable \begin{align*}x\end{align*} and \begin{align*}y\end{align*} vary inversely. Use the given \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values to write an inverse variation equation and find \begin{align*}x\end{align*} given that \begin{align*}y = 2\end{align*}.

1. \begin{align*}x=6,y=\frac{2}{3}\end{align*}
2. \begin{align*}x=16,y=\frac{3}{8}\end{align*}
3. \begin{align*}x=\frac{4}{5},y=9\end{align*}
4. \begin{align*}x=\frac{5}{6},y=\frac{18}{5}\end{align*}

Determine if the following data sets vary inversely.

1. .
\begin{align*}x\end{align*} 12 6 9 2
\begin{align*}y\end{align*} 3 6 4 18
1. .
\begin{align*}x\end{align*} 4 7 2 8
\begin{align*}y\end{align*} 10 6 20 5
1. .
\begin{align*}x\end{align*} 9 6 12 21
\begin{align*}y\end{align*} 28 42 21 12

Solve the following word problems using an inverse variation equation.

1. At a party there are 3 pizzas to share. If each pizza has 8 slices, determine how many pieces each child will receive if 12 kids attend the party. What if 8 children attend? Write an inverse variation equation to determine how many slices each child receives if there are \begin{align*}x\end{align*} kids at the party.
2. When Lionel drives from Barcelona to Madrid, 390 miles, it takes him about 6.5 hours. How fast will he have to drive in order to make the trip in 5 hours?
3. Alena and Estella can complete a job in 18 hours when they work together. If they invite Tommy to help, how long will the job take? How many friends need to work together on the job to complete it in 4 hours?
4. The temperature of the Pacific Ocean varies inversely with the depth. If the temperature at 2000 m is 2.2 degrees Celsius, what is the temperature at a depth of 4000 m?

To see the Review answers, open this PDF file and look for section 9.2.

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### Vocabulary Language: English

Constant of Proportionality

The constant of proportionality, commonly represented as $k$ is the constant ratio of two proportional quantities such as $x$ and $y$.

Direct Variation

When the dependent variable grows large or small as the independent variable does.

Inverse Variation

Inverse variation is a relationship between two variables in which the product of the two variables is equal to a constant. As one variable increases the second variable decreases proportionally.

Joint Variation

Variables exhibit joint variation if one variable varies directly as the product of two or more other variables.